On geometric bases for quantum A-polynomials of knots

Abstract

A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation theory and contact geometry. Quantization allows to present it in much simpler terms, what could make these techniques available to a broader audience. To avoid overloading of the presentation, only the case of the colored Jones polynomial for the trefoil knot is considered, though various generalizations are straightforward. Restriction to solely Jones polynomials (rather than full HOMFLY-PT) is related to a serious simplification, provided by the use of Kauffman calculus. Going beyond looks realistic, however it remains a problem, both challenging and promising.

16 pages, v2: minor updates

References in corpus (5)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• Wall Crossing Invariants: from quantum mechanics to knots
• Perspectives of differential expansion
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links

Cited by in corpus (2)

• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket